3.7.49 \(\int \frac {1}{\sqrt {x} \sqrt {1-a^2 x^2}} \, dx\) [649]

Optimal. Leaf size=21 \[ \frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right |-1\right )}{\sqrt {a}} \]

[Out]

2*EllipticF(a^(1/2)*x^(1/2),I)/a^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {335, 227} \begin {gather*} \frac {2 F\left (\left .\text {ArcSin}\left (\sqrt {a} \sqrt {x}\right )\right |-1\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 - a^2*x^2]),x]

[Out]

(2*EllipticF[ArcSin[Sqrt[a]*Sqrt[x]], -1])/Sqrt[a]

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} \sqrt {1-a^2 x^2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x^4}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 F\left (\left .\sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )\right |-1\right )}{\sqrt {a}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.02, size = 24, normalized size = 1.14 \begin {gather*} 2 \sqrt {x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};a^2 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[1 - a^2*x^2]),x]

[Out]

2*Sqrt[x]*Hypergeometric2F1[1/4, 1/2, 5/4, a^2*x^2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(15)=30\).
time = 0.12, size = 66, normalized size = 3.14

method	result	size
meijerg	\(2 \sqrt {x}\, \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], a^{2} x^{2}\right )\)	\(19\)
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x +1}\, \sqrt {-2 a x +2}\, \sqrt {-a x}\, \EllipticF \left (\sqrt {a x +1}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, a \left (a^{2} x^{2}-1\right )}\)	\(66\)
elliptic	\(\frac {\sqrt {-x \left (a^{2} x^{2}-1\right )}\, \sqrt {a \left (x +\frac {1}{a}\right )}\, \sqrt {-2 a \left (x -\frac {1}{a}\right )}\, \sqrt {-a x}\, \EllipticF \left (\sqrt {a \left (x +\frac {1}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x}\, \sqrt {-a^{2} x^{2}+1}\, a \sqrt {-a^{2} x^{3}+x}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/x^(1/2)*(-a^2*x^2+1)^(1/2)*(a*x+1)^(1/2)*(-2*a*x+2)^(1/2)*(-a*x)^(1/2)*EllipticF((a*x+1)^(1/2),1/2*2^(1/2))
/a/(a^2*x^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 0.39, size = 36, normalized size = 1.71 \begin {gather*} \frac {\sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(1/2)/(-a**2*x**2+1)**(1/2),x)

[Out]

sqrt(x)*gamma(1/4)*hyper((1/4, 1/2), (5/4,), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(5/4))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-a^2*x^2 + 1)*sqrt(x)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{\sqrt {x}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(1/(x^(1/2)*(1 - a^2*x^2)^(1/2)), x)

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